G(x1+x2)=\sum g(a_{i,j}x^ix^j)

  • Thread starter John Creighto
  • Start date
In summary, the conversation discusses the possibility of analytically finding coefficients for a function g, given a transformation that resembles a polynomial. There may be a family of functions where this is possible, but it is unclear if it has any practical applications.
  • #1
John Creighto
495
2
I wonder in what circumstances:

given a function g, we can analytically find the coefficients [tex]a_{i,j}[/tex]

[tex]g(x1+x2)=\sum a_{i,j}g(x^ix^j)[/tex]

I'm not if this would server any useful applications but the transformation looks interesting to me. It looks very simmilar to a polynomial transformation.
 
Physics news on Phys.org
  • #2
John Creighto said:
I wonder in what circumstances:

given a function g, we can analytically find the coefficients [tex]a_{i,j}[/tex]

[tex]g(x1+x2)=\sum a_{i,j}g(x^ix^j)[/tex]

I'm not if this would server any useful applications but the transformation looks interesting to me. It looks very simmilar to a polynomial transformation.

For an arbitrary function g(x), I doubt it. There might be a family of functions though.
 

Related to G(x1+x2)=\sum g(a_{i,j}x^ix^j)

1. What does "G(x1+x2)=\sum g(a_{i,j}x^ix^j)" mean?

This notation represents a function G that takes the sum of two variables, x1 and x2, and maps it to the sum of g(a_{i,j}x^ix^j) for all values of i and j. In other words, it is a function that combines the values of x1 and x2 and produces a new value based on the coefficients a_{i,j} and the powers of x.

2. How do I read this equation?

The equation can be read as "G of x1 plus x2 equals the sum of g of a sub i,j times x to the i power times x to the j power."

3. What is the purpose of using summation in this equation?

The summation symbol, represented by the Greek letter sigma, allows us to simplify the notation by indicating that the expression following it should be repeated for all values of i and j. This makes it easier to represent and manipulate functions with multiple variables and coefficients.

4. Can you give an example of how this equation is used in science?

This equation is commonly used in fields such as physics and chemistry to represent the energy or potential of a system. For example, in thermodynamics, the Gibbs free energy (G) is calculated using a similar equation, where the coefficients (a_{i,j}) represent the number of moles of each component and the powers of x represent the standard state of each component.

5. What are the limitations of this equation?

Like any mathematical equation, this equation has limitations in its applicability. It may not accurately represent certain systems or phenomena, and the coefficients and powers used in the equation may not always have a physical meaning. Additionally, this equation is limited to functions that can be represented as a sum of terms, so it may not be suitable for more complex functions.

Similar threads

MHB How to find F(x,y) for $f(x,y)= h(y)\hat{i} + g(x)\hat{j}?$
    • Calculus
Replies
11
Views
1K
I Questions about algebraic curves and homogeneous polynomial equations
    • Differential Geometry
Replies
4
Views
1K
I Nonlinear least squares vs OLS
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
586
I Fourier transform of a sum of shifted Gaussians
    • Calculus
Replies
4
Views
5K
A Multiplying divergent integrals using Hardy fields approach
    • Calculus
Replies
3
Views
1K
A Summing over continuum and uncountable numerocities
    • Calculus
Replies
1
Views
1K
Question about Spectral Estimation using AR Models
    • General Engineering
Replies
1
Views
791
Optimization methods with bivariate functions
    • Calculus
Replies
3
Views
1K
I Attempting to find an intuitive proof of the substitution formula
    • Calculus
Replies
7
Views
1K
Compact summary of Fourier series equations
    • Calculus
Replies
11
Views
901

Hot Threads

Recent Insights

Back
Top